I suppose one must say, for Descartes, the automaton was specifically mechanistic. Which is a difference.
The first section of the SEP article on Zombies, discusses the relation to Descartes' Automatons.
Descartes held that non-human animals are automata: their behavior is wholly explicable in terms of physical mechanisms. But human behavior (he argued) could not be explained in that way. Exploring the idea of a machine that would look and behave like a human being, he thought two things would unmask it: it could not use language creatively, and it could not produce appropriate non-verbal behavior in arbitrarily various situations (Discourse V). For him, therefore, no machine could behave like a human being. Knowing only seventeenth century technology, he concluded that to explain distinctively human behavior required something beyond the physical: an immaterial mind....
Notably, Descartes was a substance dualist and thought there needed to be a mental substance to make humans more than automatons.
Most modern dualists are substance monists and property dualists: they hold that there is only physical substance (usually) but that it can have both physical and mental properties. Chalmers was, I believe, an emergentist about mental properties, hold that they emerge from (supervene on) the physical properties but nevertheless cannot be reduced to them. From the IEP:
Chalmers defines a high-level phenomenon as strongly emergent when it is systematically determined by low-level facts but nevertheless truths concerning that phenomenon are in principle not deducible from truths in the lower-level domain. The question is posed by Chalmers in terms of conceptual entailment failure. That is, emergent phenomena are nomologically but not logically supervenient on lower-level facts and therefore novel fundamental laws are needed to connect properties of the two domains.
The point of Chalmers' zombie is its use in his conceivability arguments. They are supposed to be complete physical duplicates of conscious beings (people), but lacking the emergent mental properties he takes to be required for consciousness. They behave like people, talk like people, cry like people and would be indistinguishable from a conscious being, yet they lack consciousness. If such creatures are conceivable, Chalmers argues, then truths about consciousness emerge from, but aren't conceptually entailed by, truths about physical composition. Notably, he doesn't need to claim that zombies are possible in worlds with the same laws of physics as ours -- it might be that zombies are nomologically impossible. His claim is a weaker (but controversial) claim about their conceivability entailing their metaphysical possibility. Since physicalism as a metaphysical thesis about the mind is typically taken to be (metaphysically) necessarily true, if true at all, this is meant to contradict physicalism.
So, back to Descartes. Since Descartes thought that a difference in substance was required for human consciousness and distinctively human behavior, he would not have considered such Zombies to be possible. (More generally, he also seems to have held a sort of behaviorism about the mental that has fallen out of fashion since the 50's or so, and is incompatible with a Chalmers style zombie since they presuppose identical behavior with no underlying mental activity.) Plausibly, however, if he had the benefit of another couple of hundred of years of science he wouldn't discount the possibility.
TL;DR They're similar, but differ in the sort of dualism they presuppose/support.